Change-Making Problem Research Articles

On a variant of the change-making problem

The change-making problem (CMP), introduced in 1970, is a classic problem in combinatorial optimisation. It was proven to be NP-hard in 1975, but it can be solved in pseudo-polynomial time by dynamic programming. In 1999, Heipcke presented a variant of the CMP which, at first glance, looks harder than the standard version. We show that, in fact, her variant can be solved in polynomial time.

Bilateral optimal denomination problem

A common problem about currency denomination is the change-making problem. In this paper, a generalized denomination-related problem inspired by the optimal denomination problem is discussed, namely the bilateral optimal denomination problem. The solution maximizes a certain monetary value and thus can be applied in pricing strategies. Two key issues need to be addressed. One is how to properly formulate the problem in mathematical language. The other is how to compute the corresponding result with an algorithm. The paper learns from existing denomination algorithms and provides a solution to the problem from construction to implementation. Some limitations are needed to improve the practicality of the solution. An example of threshold discounts is used to illustrate the real-life application of the proposed problem.

Characterization of canonical systems with six types of coins for the change-making problem

This paper analyzes a necessary and sufficient condition for the change-making problem to be solvable with a greedy algorithm. The change-making problem is to minimize the number of coins used to pay a given value in a specified currency system. This problem is NP-hard, and therefore the greedy algorithm does not always yield an optimal solution. Yet for almost all real currency systems, the greedy algorithm outputs an optimal solution. A currency system for which the greedy algorithm returns an optimal solution for any value of payment is called a canonical system. Canonical systems with at most five types of coins have been characterized in previous studies. In this paper, we give characterization of canonical systems with six types of coins, and we propose a partial generalization of characterization of canonical systems.

More on change-making and related problems

Given a set of n integer-valued coin types and a target value t, the well-known change-making problem asks for the minimum number of coins that sum to t, assuming an unlimited number of coins in each type. In the more general all-targets version of the problem, we want the minimum number of coins summing to j, for every j=0,…,t. We obtain a number of new results on these problems, including (i) an algorithm for the all-targets problem with running time O˜(t4/3) (improving a previous O˜(t3/2)-time algorithm), (ii) a very simple O˜(u2+t)-time algorithm for the all-targets problem, where u denotes the maximum coin value, and (iii) an algorithm for the original (single-target) problem with running time O˜(u) (improving a previous O˜(t)-time algorithm by Chan and He (SOSA'20)). We also describe applications to the (single-capacity) unbounded knapsack problem and the minimum word break problem.

Change-making problems revisited: a parameterized point of view

The change-making problem is the problem of representing a given amount of money with the fewest number of coins possible from a given set of coin denominations. In the general version of the problem, an upper bound for the availability of every coin value is given. Even the special case, where for each value an unlimited number of coins is available, is NP-hard. Since in the original problem some amounts can not be represented, especially if no coin of value one exists, we introduce generalized problems that look for approximations of the given amount such that a cost function is minimized. We recall algorithms for the change-making problem and present new algorithms for the generalized version of the problem. Motivated by the NP-hardness we study fixed-parameter tractability of all these problems. We show that some of these problems are fixed-parameter tractable and that some are $$\hbox {W}[1]$$ -hard. In order to show the existence of polynomial and constant-size kernels we prove some general results and apply them to several parameterizations of the change-making problems.

Combinatorics of the change-making problem

We investigate the structure of the currencies (systems of coins) for which the greedy change-making algorithm always finds an optimal solution (that is, a one with minimum number of coins). We present a series of necessary conditions that must be satisfied by the values of coins in such systems. We also uncover some relations between such currencies and their sub-currencies.

A polynomial-time algorithm for the change-making problem

Optimally making change—representing a given value with the fewest coins from a set of denominations—is in general NP-hard. In most real money systems however, the greedy algorithm is optimal. We give a polynomial-time algorithm to determine, for a given coin system, whether the greedy algorithm is optimal.

Optimal bounds for the change-making problem

The change-making problem is the problem of representing a given value with the fewest coins possible. We investigate the problem of determining whether the greedy algorithm produces an optimal representation of all amounts for a given set of coin denominations 1= c 1< c 2<⋯< c m . Chang and Gill (1970) show that if the greedy algorithm is not always optimal, then there exists a counterexample x in the range c 3≤x< c m(c mc m−1+c m−3c m−1) c m−c m−1 . To test for the existence of such a counterexample, Chang and Gill propose computing and comparing the greedy and optimal representations of all x in this range. In this paper we show that if a counterexample exists, then the smallest one lies in the range c 3+1 <x<c m+c m−1 .

Optimal Bounds for the Change-Making Problem

&lt;p&gt;The change-making problem is the problem of representing a given value with the fewest coins possible. We investigate the problem of determining whether the greedy algorithm produces an optimal representation of all amounts for a given set of coin denominations 1 = c_1 &amp;lt; c_2 &amp;lt; ... &amp;lt; c_m. Chang and Gill show that if the greedy algorithm is not always optimal, then there exists a counterexample x in the range&lt;/p&gt;&lt;p&gt;c_3 &amp;lt;= x &amp;lt; (c_m(c_m c_m-1 + c_m - 3c_m-1)) \ (c_m - c_m-1).&lt;/p&gt;&lt;p&gt;To test for the existence of such a counterexample, Chang and Gill propose computing and comparing the greedy and optimal representations of all x in this range. In this paper we show that if a counterexample exists, then the smallest one lies in the range c_3 + 1 &amp;lt; x &amp;lt; c_m + c_m-1, and these bounds are tight. Moreover, we give a simple test for the existence of a counterexample that does not require the calculation of optimal representations.&lt;/p&gt;&lt;p&gt;In addition, we give a complete characterization of three-coin systems and an efficient algorithm for all systems with a fixed number of coins. Finally, we show that a related problem is &lt;em&gt;co&lt;/em&gt;NP-complete.&lt;/p&gt;

Knapsack Problems: Algorithms and Computer Implementations

Introduction knapsack problem bounded knapsack problem subset-sum problem change-making problem multiple knapsack problem generalized assignment problem bin packing problem. Appendix: computer codes.

Optimal and canonical solutions of the change making problem

Given a set of different coin-types, the change-making problem consists in assembling a changes c so as to minimize the number of coins utilized; this problem can arise in practice in some classes of uni-dimensional cargo-loading and cutting-stock problems. The paper presents a depth-first branch and bound algorithm, which makes use of a new dominance criterion; the superiority of this algorithm over Chang-Gill's and Wright's is shown through extensive computational comparisons. Moreover the performance and the applicability of a heuristic algorithm are analyzed.

The Change-Making Problem

The Change-Making Problem

Algorithmic Solution of the Change-Making Problem

Algorithmic Solution of the Change-Making Problem